On some polynomials enumerating Fully Packed Loop configurations: Evaluation at negative values

When considering matchings with p nested arches these quantities are known to be polynomials. In a recent article, Fonseca and Nadeau conjectured some unexpected properties of these polynomials, suggesting that these quantities could be combinatorially interpreted even for negative p. Here, we prove some conjectures in this article. Notably, we prove that for negative p we can factor the polynomials into two parts a “positive” one and a “negative” one. Also, a sum rule of the negative part is proven here.